Theorem en2other2 8715

Description: Taking the other element twice in a pair gets back to the original element. (Contributed by Stefan O'Rear, 22-Aug-2015.)

Assertion

Ref	Expression
en2other2	⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → ∪ (𝑃 ∖ {∪ (𝑃 ∖ {𝑋})}) = 𝑋)

Proof of Theorem en2other2

Step	Hyp	Ref	Expression
1		en2eleq 8714	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → 𝑃 = {𝑋, ∪ (𝑃 ∖ {𝑋})})
2		prcom 4211	. . . . . . 7 ⊢ {𝑋, ∪ (𝑃 ∖ {𝑋})} = {∪ (𝑃 ∖ {𝑋}), 𝑋}
3	1, 2	syl6eq 2660	. . . . . 6 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → 𝑃 = {∪ (𝑃 ∖ {𝑋}), 𝑋})
4	3	difeq1d 3689	. . . . 5 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → (𝑃 ∖ {∪ (𝑃 ∖ {𝑋})}) = ({∪ (𝑃 ∖ {𝑋}), 𝑋} ∖ {∪ (𝑃 ∖ {𝑋})}))
5		difprsnss 4270	. . . . 5 ⊢ ({∪ (𝑃 ∖ {𝑋}), 𝑋} ∖ {∪ (𝑃 ∖ {𝑋})}) ⊆ {𝑋}
6	4, 5	syl6eqss 3618	. . . 4 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → (𝑃 ∖ {∪ (𝑃 ∖ {𝑋})}) ⊆ {𝑋})
7		simpl 472	. . . . . 6 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → 𝑋 ∈ 𝑃)
8		1onn 7606	. . . . . . . . . 10 ⊢ 1𝑜 ∈ ω
9	8	a1i 11	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → 1𝑜 ∈ ω)
10		simpr 476	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → 𝑃 ≈ 2𝑜)
11		df-2o 7448	. . . . . . . . . 10 ⊢ 2𝑜 = suc 1𝑜
12	10, 11	syl6breq 4624	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → 𝑃 ≈ suc 1𝑜)
13		dif1en 8078	. . . . . . . . 9 ⊢ ((1𝑜 ∈ ω ∧ 𝑃 ≈ suc 1𝑜 ∧ 𝑋 ∈ 𝑃) → (𝑃 ∖ {𝑋}) ≈ 1𝑜)
14	9, 12, 7, 13	syl3anc 1318	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → (𝑃 ∖ {𝑋}) ≈ 1𝑜)
15		en1uniel 7914	. . . . . . . 8 ⊢ ((𝑃 ∖ {𝑋}) ≈ 1𝑜 → ∪ (𝑃 ∖ {𝑋}) ∈ (𝑃 ∖ {𝑋}))
16		eldifsni 4261	. . . . . . . 8 ⊢ (∪ (𝑃 ∖ {𝑋}) ∈ (𝑃 ∖ {𝑋}) → ∪ (𝑃 ∖ {𝑋}) ≠ 𝑋)
17	14, 15, 16	3syl 18	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → ∪ (𝑃 ∖ {𝑋}) ≠ 𝑋)
18	17	necomd 2837	. . . . . 6 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → 𝑋 ≠ ∪ (𝑃 ∖ {𝑋}))
19		eldifsn 4260	. . . . . 6 ⊢ (𝑋 ∈ (𝑃 ∖ {∪ (𝑃 ∖ {𝑋})}) ↔ (𝑋 ∈ 𝑃 ∧ 𝑋 ≠ ∪ (𝑃 ∖ {𝑋})))
20	7, 18, 19	sylanbrc 695	. . . . 5 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → 𝑋 ∈ (𝑃 ∖ {∪ (𝑃 ∖ {𝑋})}))
21	20	snssd 4281	. . . 4 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → {𝑋} ⊆ (𝑃 ∖ {∪ (𝑃 ∖ {𝑋})}))
22	6, 21	eqssd 3585	. . 3 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → (𝑃 ∖ {∪ (𝑃 ∖ {𝑋})}) = {𝑋})
23	22	unieqd 4382	. 2 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → ∪ (𝑃 ∖ {∪ (𝑃 ∖ {𝑋})}) = ∪ {𝑋})
24		unisng 4388	. . 3 ⊢ (𝑋 ∈ 𝑃 → ∪ {𝑋} = 𝑋)
25	24	adantr 480	. 2 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → ∪ {𝑋} = 𝑋)
26	23, 25	eqtrd 2644	1 ⊢ ((𝑋 ∈ 𝑃 ∧ 𝑃 ≈ 2𝑜) → ∪ (𝑃 ∖ {∪ (𝑃 ∖ {𝑋})}) = 𝑋)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   ∖ cdif 3537  {csn 4125  {cpr 4127  ∪ cuni 4372   class class class wbr 4583  suc csuc 5642  ωcom 6957  1_𝑜c1o 7440  2_𝑜c2o 7441   ≈ cen 7838

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-1o 7447  df-2o 7448  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845

This theorem is referenced by:  pmtrfinv  17704